arXiv subjects

Timothy E. Holy

Publications and source records attributed to Timothy E. Holy.

5 recordsLinked to original sources

arXiv2026

Jordan algebras, hemiplex numbers, and the Cholesky decomposition of arbitrary symmetric matrices

Positive-semidefinite matrices are most efficiently factored using the Cholesky decomposition. For indefinite matrices, the Cholesky factorization does not exist, and the alternatives face greater challenges in achieving numeric stability and preservation of banded structure. Here we pursue an analogy between the requirement for positive-semidefinite matrices and the solution of the quadratic equation x^2 = c for c <= 0. It is shown that a non-associative algebra, called the hemiplex numbers, allows the Cholesky factorization to be computed for arbitrary symmetric matrices. Crucially, the hemiplex Cholesky factorization does not require pivoting for its existence or stability, allowing it to preserve banded structure. For singular matrices it produces a parametrization of the null space, and provides opportunity for truncation of nearly-null directions in a manner similar to common usage of the singular value decomposition. The hemiplex Cholesky factorization may be a practically useful addition to the tools for solving symmetric linear equations.

math.NA↗

arXiv2024

An optimal pairwise merge algorithm improves the quality and consistency of nonnegative matrix factorization

Non-negative matrix factorization (NMF) is a key technique for feature extraction and widely used in source separation. However, existing algorithms may converge to poor local minima, or to one of several minima with similar objective value but differing feature parametrizations. Here we show that some of these weaknesses may be mitigated by performing NMF in a higher-dimensional feature space and then iteratively combining components with an analytically-solvable pairwise merge strategy. Experimental results demonstrate our method helps non-ideal NMF solutions escape to better local optima and achieve greater consistency of the solutions. Despite these extra steps, our approach exhibits similar computational performance to established methods by reducing the occurrence of "plateau phenomenon" near saddle points. Moreover, the results also illustrate that our method is compatible with different NMF algorithms. Thus, this can be recommended as a preferred approach for most applications of NMF.

cs.LG↗

arXiv2024

GSVD-NMF: Recovering Missing Features in Non-negative Matrix Factorization

Non-negative matrix factorization (NMF) is an important tool in signal processing and widely used to separate mixed sources into their components. Algorithms for NMF require that the user choose the number of components in advance, and if the results are unsatisfying one typically needs to start again with a different number of components. To make NMF more interactive and incremental, here we introduce GSVD-NMF, a method that proposes new components based on the generalized singular value decomposition (GSVD) to address discrepancies between the initial under-complete NMF results and the SVD of the original matrix. Simulation and experimental results demonstrate that GSVD-NMF often effectively recovers multiple missing components in under-complete NMF, with the recovered NMF solutions frequently reaching better local optima. The results further show that GSVD-NMF is compatible with various NMF algorithms and that directly augmenting components is more efficient than rerunning NMF from scratch with additional components. By deliberately starting from under-complete NMF, GSVD-NMF has the potential to be a recommended approach for a range of general NMF applications.

cs.LG↗

arXiv2021

Julia for Biologists

Increasing emphasis on data and quantitative methods in the biomedical sciences is making biological research more computational. Collecting, curating, processing, and analysing large genomic and imaging data sets poses major computational challenges, as does simulating larger and more realistic models in systems biology. Here we discuss how a relative newcomer among computer programming languages -- Julia -- is poised to meet the current and emerging demands in the computational biosciences, and beyond. Speed, flexibility, a thriving package ecosystem, and readability are major factors that make high-performance computing and data analysis available to an unprecedented degree to "gifted amateurs". We highlight how Julia's design is already enabling new ways of analysing biological data and systems, and we provide a, necessarily incomplete, list of resources that can facilitate the transition into the Julian way of computing.

q-bio.QM↗

arXiv1997

The Analysis of Data from Continuous Probability Distributions

Conventional statistics begins with a model, and assigns a likelihood of obtaining any particular set of data. The opposite approach, beginning with the data and assigning a likelihood to any particular model, is explored here for the case of points drawn randomly from a continuous probability distribution. A scalar field theory is used to assign a likelihood over the space of probability distributions. The most likely distribution may be calculated, providing an estimate of the underlying distribution and a convenient graphical representation of the raw data. Fluctuations around this maximum likelihood estimate are characterized by a robust measure of goodness-of-fit. Its distribution may be calculated by integrating over fluctuations. The resulting method of data analysis has some advantages over conventional approaches.

physics.data-an↗